Biology & Bayes in Exposure-Response

The next generation of model averaged benchmark exposures is likely to provide enhanced capabilities for expressing plausible model shapes, in terms of “shape features.”  These would not necessarily be the parameters chosen to express a given model.  A shape feature would be interpretable in the same way for each model.  A general computational approach is briefly mentioned.

Consistently with the general emphasis of the blog, ideas are solicited on how to represent the prior plausibility of general curve shapes, from a standpoint of biological insight or experience.

For a given dataset, different dose-response models will tend to yield somewhat different estimates and bounds for benchmark exposure. A final value may be based on selection of one model or on model averaging. Either way, the prior plausibility of given models for a given context (endpoint and substance class) is ideally taken into account, as well as agreement of individual models with data modeled. (I have, however, some frequentist- minded reservations about frequencies of parameter values in historical data). The model averaged bound computed with the current generation of methods is a percentile of a parameter uncertainty distribution, which represents the relatively plausibility of alternative choices of the benchmark exposure, considering both uncertainty in model choice, and uncertainty of model parameters. Let us be clear: no-one is talking about a simple averaging of point estimates (or respective bounds) from individual models. Point estimates are not actually required. (The Bayesian-informed will recognize our uncertainty distributions as posterior distributions.)The combined uncertainty distribution can be computed in three steps. First, an uncertainty distribution for the benchmark exposure is computed separately for each model, assuming it to be the true model. Second we compute, for each model a posterior weight, which is the probability that it is the true model, in light of data, (We proceed as if the true model is one of a specified, finite set.) Finally, the posterior weights are used in Bayes theorem to combine the model-specific distributions.

The second step requires a prior weight for each model, the probability we would give the model based on biological plausibility or experience, without seeing the data. The easiest approach is to assign equal prior probability to model. However, I doubt the models will be seen as exhausting the biological hypotheses. (If they could, we still might view some as more plausible than others.) A given curve can take on a diversity of shapes based on different choices of values for model parameters. With equal prior weights it is difficult to know exactly what is being assumed about plausible curve shapes. Can we do better than equal weights?

However, I scarcely think the following sort of possibility has gone unnoticed up to now: Instead of assigning prior probability to the set adopted for some software package, prior probability could be assigned to qualitatively distinct curve shapes such as convex of concave. To accomplish this it seems we would get away from working with the parameters that have been chosen to represent a given model, and work with descriptors of shape that would have the same interpretation across models. For example, every continuous, increasing function would have a number of inflection points, perhaps zero, so the number of inflection points is an example of a shape feature. (“Inflection point” is used following its proper calculus definition, particularly not to mean an x value where a curve starts to increase rapidly.)

Now for some implementation issues. Sometimes models may usefully be re-expressed using parameters that can serve as shape features. I doubt it will be productive to try to rely largely on re-parametrizations. I think one might work with inconveniently-parameterized models and post-process the posterior samples using importance sampling. I think the further details can be developed based on basic Bayesian computational methodology. However, I will say that the approach seems to require separate sampling of the prior, to determine prior probabilities of curve categories defined in terms of shape features.